Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 11 (2006) > Paper 48 open journal systems 


Gaussian fluctuations in complex sample covariance matrices

Zhonggen Su, Zhejiang University


Abstract
Let X=(Xi,j)m×n, m ≥ n, be a complex Gaussian random matrix with mean zero and variance 1/n, let S=X*X be a sample covariance matrix. In this paper we are mainly interested in the limiting behavior of eigenvalues when m/n -> γ ≥ 1. Under certain conditions on k, we prove the central limit theorem holds true for the k-th largest eigenvalues l(k) as k tends to infinity as n->∞. The proof is largely based on the Costin-Lebowitz-Soshnikov argument and the asymptotic estimates for the expectation and variance of the number of eigenvalues in an interval. The standard technique for the RH problem is used to compute the exact formula and asymptotic properties for the mean density of eigenvalues. As a by-product, we obtain a convergence speed of the mean density of eigenvalues to the Marchenko-Pastur distribution density under the condition | m/n-g|=O(1/n)


Full text: PDF

Pages: 1284-1320

Published on: December 17, 2006
Bibliography
  1. Borodin, Alexei; Okounkov, Andrei; Olshanski, Grigori. Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 (2000), no. 3, 481--515 (electronic). MR1758751 (2001g:05103)
  2. bibitem{CL1995} O. Costin, J. Lebowitz, {Gaussian fluctuations in random matrices. } {em Phys. Rev. Lett. }, (1995) {bf 75} (1), 69-72.
  3. Deift, P. A. Orthogonal polynomials and random matrices: a Riemann-Hilbert York; American Mathematical Society, Providence, RI, 1999. viii+273 pp. ISBN: 0-9658703-2-4; 0-8218-2695-6 MR1677884 (2000g:47048)
  4. Ercolani, N. M.; McLaughlin, K. D. T.-R. Asymptotics of the partition function for random matrices via Int. Math. Res. Not. 2003, no. 14, 755--820. MR1953782 (2005f:82048)
  5. Goodman, N. R. Statistical analysis based on a certain multivariate complex Gaussian Ann. Math. Statist. 34 1963 152--177. MR0145618 (26 #3148a)
  6. Götze, Friedrich; Tikhomirov, Alexander. The rate of convergence for spectra of GUE and LUE matrix Cent. Eur. J. Math. 3 (2005), no. 4, 666--704 (electronic). MR2171668 (2006j:60022)
  7. Gustavsson, Jonas. Gaussian fluctuations of eigenvalues in the GUE. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 2, 151--178. MR2124079 (2005k:60074)
  8. Johansson, Kurt. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), no. 2, 437--476. MR1737991 (2001h:60177)
  9. Khatri, C. G. Classical statistical analysis based on a certain multivariate complex Ann. Math. Statist. 36 1965 98--114. MR0192598 (33 #823)
  10. König, Wolfgang. Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2 (2005), 385--447 (electronic). MR2203677
  11. Marv cenko, V. A.; Pastur, L. A. Distribution of eigenvalues in certain sets of random matrices. (Russian) Mat. Sb. (N.S.) 72 (114) 1967 507--536. MR0208649 (34 #8458)
  12. Olver, Frank W. J. Asymptotics and special functions. AKP Classics. A K Peters, Ltd., Wellesley, MA, 1997. xviii+572 pp. ISBN: 1-56881-069-5 MR1429619 (97i:41001)
  13. Soshnikov, Alexander B. Gaussian fluctuation for the number of particles in Airy, Bessel, sine, J. Statist. Phys. 100 (2000), no. 3-4, 491--522. MR1788476 (2001m:82006)
  14. M. Vanlessen, Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory. arXiv:math.CA/0504604 v2 (2005).
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489